Find the average rate of change of f(x)=√x from x1=4 to x2=9.

Ch. 3 - Polynomial and Rational Functions

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Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

문제 100

4장, 문제 100

Find the average rate of change of f(x)=√x from x₁=4 to x₂=9.

검증된 단계별 안내

Identify the function and the interval: The function is

f (x) = \sqrt{x}

, and the interval is from

x

₁ = 4 to

x

₂ = 9.

Recall the formula for the average rate of change of a function

f

over an interval

[x_1, x_2]

\(\frac{f(x_2) - f(x_1)}{x_2 - x_1}\)

Calculate the function values at the endpoints: find

f(4) = \(\sqrt{4}\)

and

f(9) = \(\sqrt{9}\)

Substitute the values into the average rate of change formula:

\(\frac{f(9) - f(4)}{9 - 4}\)

Simplify the expression by evaluating the square roots and performing the subtraction and division to find the average rate of change.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Average Rate of Change

The average rate of change of a function between two points measures how much the function's output changes per unit change in input. It is calculated as the difference in function values divided by the difference in input values, similar to the slope of a secant line connecting the two points on the graph.

Square Root Function

The square root function, denoted as f(x) = √x, outputs the non-negative number whose square is x. It is defined for x ≥ 0 and is a nonlinear function that increases at a decreasing rate, meaning its graph is a curve that flattens as x increases.

Evaluating Function Values

To find the average rate of change, you must evaluate the function at the given points x1 and x2. This involves substituting the x-values into the function to find f(x1) and f(x2), which are then used to compute the difference quotient.